Inequality and information are concepts from largely different communities. Information theory originated as a mathematical tool to study communication systems, while inequality theory evolved in economics and sociology. These areas developed measures to provide important quantitative insights for their respective applications. However, both domains also need the ability to decompose their measures to shed light on the structure of inequality within a population, or to understand how different pieces of information are provided and interact. 

In 2010, researchers in information theory developed a technique to decompose how information is provided redundantly by multiple sources, uniquely by a particular source, and synergistically by the interactions between sources. The proposed framework found favor in the community, but the decomposition measure was strongly criticized. Despite significant research efforts, no suitable replacement could be found. The primary contribution of this work is a solution to this open question: We developed a decomposition measure to non-negatively quantify the partial contributions of an arbitrary number of sources about a target with practical operational interpretation. 

Surprisingly, the underlying representation we used for our decomposing is equivalent to the representation for quantifying inequality. This identified relationship between measures of information and inequality enabled the direct transfer of results. Consequently, we generalized established inequality measures into a new family and provided a novel decomposition that characterizes inequality by the redundant, unique, and synergetic interactions between attributes of individuals. Finally, we demonstrate that subgroup decompositions from inequality theory highlight a recursive subgroup structure of information measures. 

The developed techniques can directly provide novel insights for studying applied questions in complex systems, information theory, economics, and sociology. Our tools enable professionals to gain deeper insights, understand the impact of changes and make informed decisions.